Transposition bijects even and odd permutations

Dependencies:

1. Parity of a permutation

Let ${\lambda }_{\tau }(\sigma )=\tau \sigma$, where $\tau$ is a transposition.

${\lambda }_{\tau }$ is a bijection from the set of even permutations to the set of odd permutations.

Proof

$\text{Let }{\lambda }_{\tau }({\sigma }_1)={\lambda }_{\tau }({\sigma }_2)⟹\tau {\sigma }_1=\tau {\sigma }_2⟹{\sigma }_1={\tau }^{-1}\tau {\sigma }_2={\sigma }_2$

Therefore, ${\lambda }_{\tau }$ is one-to-one.

Let $\mu$ be an odd permutation. Then ${\lambda }_{\tau }(\mu )=\tau \mu$ is an even permutation.

${\lambda }_{\tau }({\lambda }_{\tau }(\mu ))=\tau \tau \mu =\mu$. So $\mu$ has an even permutation as a pre-image in ${\lambda }_{\tau }$. Therefore, ${\lambda }_{\tau }$ is onto.

Therefore, ${\lambda }_{\tau }$ is a bijection from even permutations to odd permutations.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 10

Transitive dependencies:

1. /sets-and-relations/relation-composition-is-associative
2. /sets-and-relations/composition-of-bijections-is-a-bijection
3. Group
4. Subgroup
5. Permutation group
6. Product of cycles and a transposition
7. Permutation is disjoint cycle product
8. Product of disjoint cycles is commutative
9. Canonical cycle notation of a permutation
10. Parity of a permutation